We would like to know more generally what states in \(\mathcal {H}_L \otimes \mathcal {H}_R\)  can have such a wormhole description. Entanglement is clearly necessary (in its absence, correlations \(\langle \mathcal {O}_L \mathcal {O}_R \rangle \)  in non-interacting systems would vanish), but it has been argued that it is not sufficient. In [1]}, [2]}, it was argued that in a generic state with the same degree of entanglement as the TFD, the correlations of simple operators between the two boundaries is exponentially small, so generic entangled states do not have a geometric wormhole description. Some studies have considered small departures from the TFD, finding that these lead to longer wormholes. This was first explored in [3]}, who considered a local operator insertion on the TFD evolved in Lorentzian time. A related investigation in [4]} considered operator insertions in the Euclidean path integral defining the TFD state. We will review these studies in the next section.
